228 



PIERCE. 



By this proposition, referring to Figure 10, we see that 



IT 



TT 



COS \p = COS - COS ^ + sin - sin 6 cos 

 = sin 6 cos 4>. 



(85) 



Figure 10. 



By the same proposition 



or 



cos X = cos d cos ^ + sin ^ sin \f/ cos a; 



cos a = — 



cos ^ cos i/' 

 sin sin^ '/' 



cos a 

 sini/' 



and by (85) this becomes 



cos a 

 sin;/' 



cos 6 cos ;/' 

 sin sin^ i/' ' 



cos cos <f) 

 1 — sin^^cos^<> 



(86) 



(87) 



(88) 



17. Integration for Mutual Power. — Now substituting the 

 trigonometric relations (83), (84), (85), (88) into equation (82), we 

 obtain the following integral expression for the time average of the 

 mutual power radiated through the aerial hemisphere-. 



